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Function and Operator Theory on Large Bergman Spaces

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Function and Operator Theory on Large Bergman Spaces View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Diposit Digital de la Universitat de Barcelona Function and Operator Theory on Large Bergman spaces Hicham Arroussi A questa tesi doctoral està subjecta a la llicència Reconeixement 3.0. Es panya de Creative Commons . Esta tesis doctoral está sujeta a la licencia Reconocimi ento 3.0. España de Creative Commons . Th is doctoral thesis is licensed under the Creative Commons Att ribution 3.0. Spain License . Function and Operator Theory on Large Bergman spaces Hicham Arroussi Programa de Doctorat en Matem`atiquesi Inform`atica Universitat of Barcelona A thesis submitted for the degree of Doctor in Mathematics (PhD) Advisor: Dr. Jordi Pau Plana Declaration Hicham Arroussi wrote this thesis for the award of Doctor of Mathematics under the supervision of Jordi Pau Plana, and Joaquim Ortega-Cerd`aas a tutor. Dr. Jordi Pau Plana Barcelona, February 2016 iii iv Acknowledgements First, I would like to express my sincerest thanks to my supervisor Dr. Jordi Pau. Through- out this project his enthusiasm for the subject matter was a great source of inspiration. In addition to this, his excellent supervision kept me on track yet still allowed me to produce my best work. I must also thank the staff of the Department of Applied Mathematics and Analysis who have made this time so enjoyable and for being there whenever I needed help, especially Joaquim Ortega-Cerd`a,Xavier Massaneda, Daniel Pascuas, Joan F`abrega,Fina Casasayas, Maria J. Carro, Javier Soria, Oriol Pujol, and Konstantin Dyakonov with whom I have shared the office. I am also extremely grateful to Josep Vives from the Department of Probability, Logic and Statistics. All of them were available when I wanted to talk maths as well as those times when I would have preferred to speak about anything else. To my close friends and family I will be eternally grateful. Without your encourage- ment I would never have completed this work. You will never know how much your love, kindness and support has meant to me these past years and I just hope that I can pay it all back in the years to come. v vi Contents Introduction 1 1 Large Bergman spaces 5 1.1 Basic properties . 5 1.2 The class of weights W and examples . 12 1.3 Test functions and some estimates . 13 1.4 Carleson type measures . 21 1.5 Atomic Decomposition . 23 2 Toeplitz operators 29 2.1 Boundedness and compactness . 30 2.2 Membership in Schatten classes . 32 3 Area operators 45 3.1 The case p ≤ q ................................. 46 3.2 The case q < p ................................. 49 4 The exponential weight 53 4.1 Integral estimates for reproducing kernels . 53 4.2 Bounded projections . 54 4.3 Complex interpolation and duality . 57 4.4 Atomic decomposition . 62 4.5 Toeplitz operators . 64 4.6 Hankel operators . 73 4.7 Examples of weights in the class E ....................... 80 5 Open questions and future research 97 5.1 Extension of the results to p 6=2........................ 97 5.2 Localization and Compactness . 98 5.3 Big Hankel operators . 100 5.4 Small Hankel operators . 101 Bibliography 103 vii viii Introduction The theory of Bergman spaces has been a central subject of study in complex analysis during the past decades. The book [7] by S. Bergman contains the first systematic treat- ment of the Hilbert space of square integrable analytic functions with respect to Lebesgue area measure on a domain. His approach was based on a reproducing kernel that became known as the Bergman kernel function. When attention was later directed to the spaces Ap over the unit disk, it was natural to call them Bergman spaces. As counterparts of Hardy spaces, they presented analogous problems. However, although many problems in Hardy spaces were well understood by the 1970s, their counterparts for Bergman spaces were generally viewed as intractable, and only some isolated progress was done. The 1980s saw the emerging of operator theoretic studies related to Bergman spaces with important contributions by several authors. Their achievements on Bergman spaces with standard weights are presented in Zhu's book [77]. The main breakthroughs came in the 1990s, where in a flurry of important advances, problems previously considered intractable began to be solved. First came Hedenmalm`s construction of canonical divisors [26], then Seip's description [59] of sampling and interpolating sequences on Bergman spaces, and later on, the study of Aleman, Richter and Sundberg [1] on the invariant subspaces of A2, among others. This attracted other workers to the field and inspired a period of intense research on Bergman spaces and related topics. Nowadays there are rich theories on Bergman spaces that can be found on the textbooks [27] and [22]. Meanwhile, also in the nineties, some isolated problems on Bergman spaces with ex- ponential type weights began to be studied. These spaces are large in the sense that they contain all the Bergman spaces with standard weights, and their study presented new dif- ficulties, as the techniques and ideas that led to success when working on the analogous problems for standard Bergman spaces, failed to work on that context. It is the main goal of this work to do a deep study of the function theoretic properties of such spaces, as well as of some operators acting on them. It turns out that large Bergman spaces are close in spirit to Fock spaces [79], and many times mixing classical techniques from both Bergman and Fock spaces in an appropriate way, can led to some success when studying large Bergman spaces. This dissertation is structured into five chapters. 1 In Chapter 1, we give some basic properties of the large Bergman spaces Ap(!), for weights ! in a certain class W of rapidly decreasing weights, considered previously in [23] and [51]. The concrete definition of the class is given, and it is seen that contains the family of exponential type weights −A ! (z) = exp ; σ > 0; A > 0; (0.1) σ (1 − jzj2)σ and also weights decreasing even faster such as some double exponential weights. We collect some of the well known results proved in earlier works (pointwise estimates, completeness, construction of suitable test functions, description of Carleson-type measures, etc.), but we also include some new ones as the atomic decomposition for the Hilbert space A2(!), or the estimates for the solutions of the d-bar equation obtained in Theorem 1.4, a result used later in Chapter 4 in order to characterize bounded and compact Hankel operators with conjugate analytic symbols. The test functions given in Lemma C are useful in order to study several problems on Ap(!) even for p 6= 2 (see [9] for the study of sampling and interpolating sequences, or [51] for the study of the boundedness of certain Cesaro type integration operators acting between large weighted Bergman spaces), but the fact that estimates for the norm of the reproducing kernels Kz are only available when p = 2 allows us to consider other problems only in the Hilbert space case, as for example, the study of Toeplitz operators in the next chapter. The topics of this chapter will serve as a basis for our later chapters. Chapter 2 is devoted to characterize the boundedness, compactness and membership 2 in Schatten classes of the Toeplitz operator Tµ acting on A (!) for weights in our class W. If µ is a finite positive Borel measue on D, the Toeplitz operator Tµ is the integral type operator given by Z Tµf(z) = f(ξ)Kz(ξ) !(ξ) dµ(ξ); f 2 H(D); D where Kz denotes the reproducing kernel at the point z 2 D. The operator Tµ acting on standard weighted Bergman spaces has been extensively studied [75]. Luecking was probably one of the first who considered Tµ with measures as symbols, and the study of Toeplitz operators acting on large weighted Bergman spaces was initiated by Lin and Rochberg [35], where they proved descriptions of the boundedness and compactness of the Toeplitz operator in terms of the behavior of a certain averaging function of µ. Since their class of weights is slightly different of our class, we will offer a proof for the both descriptions in Theorem 2.1. The main result of this chapter is Theorem 2.8 where a complete description, valid for all 2 0 < p < 1, of when the Toeplitz operator Tµ acting on A (!) belongs to the Schatten ideal Sp. This extends the description obtained for standard Bergman spaces [78], and solves a problem posed by Lin and Rochberg in [32], where only one implication was proved. Just to mention here that the results of this chapter has been recently published in our paper [4]. 2 In Chapter 3 we study the area operator Aµ acting on large weighted Bergman spaces. Given a positive Borel measure µ on the unit disk D, the area operator Aµ is the sublinear operator defined by Z dµ(z) Aµ(f)(ζ) = jf(z)j 2 ; ζ 2 T := @D; Γ(ζ) 1 − jzj where Γ(ζ) is a typical non-tangential approach region (Stolz region) with vertex ζ 2 T. The area operator is useful in harmonic analysis, and is closely related to, for ex- ample, non-tangential maximal functions, Poisson integrals, Littlewood-Paley operators, tent spaces, etc. The study of the area operator acting on the classical Hardy spaces Hp was initiated by W. Cohn [14]. He proved that, for 0 < p < 1, the area operator p p Aµ : H ! L (T) is bounded if and only if µ is a classical Carleson measure.
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